The resultant of `vec(A)` and `vec(B)` makes an angle `alpha` with `vec(A)` and `omega` with `vec(B)` , then:-

Options

A. `alpha lt beta`
B. `alpha lt beta` if `A lt B`
C. `alpha lt beta` if `A gt B`
D. `alpha lt beta` if `A = B`

Answer

To solve the problem regarding the angles made by the resultant vector with vectors A and B, we can follow these steps:

### Step 1: Understand the Problem
We have two vectors, \(\vec{A}\) and \(\vec{B}\), and their resultant vector \(\vec{R}\) makes an angle \(\alpha\) with \(\vec{A}\) and an angle \(\beta\) with \(\vec{B}\). We need to analyze the relationship between these angles based on the magnitudes of the vectors.

### Step 2: Analyze the Angles
The resultant vector \(\vec{R}\) will be more inclined towards the vector that has a greater magnitude. This means:
- If \(|\vec{A}| < |\vec{B}|\), then \(\vec{R}\) will be closer to \(\vec{B}\) and \(\beta\) will be smaller than \(\alpha\).
- If \(|\vec{A}| > |\vec{B}|\), then \(\vec{R}\) will be closer to \(\vec{A}\) and \(\alpha\) will be smaller than \(\beta\).
- If \(|\vec{A}| = |\vec{B}|\), then \(\alpha\) will equal \(\beta\).

### Step 3: Establish Relationships
From the above analysis:
1. If \(|\vec{A}| < |\vec{B}|\), then \(\beta < \alpha\).
2. If \(|\vec{A}| > |\vec{B}|\), then \(\alpha < \beta\).
3. If \(|\vec{A}| = |\vec{B}|\), then \(\alpha = \beta\).

### Step 4: Evaluate the Options
Now, we can evaluate the given options based on the relationships established:
- **Option 1**: If \(|\vec{A}| < |\vec{B}|\), then \(\beta < \alpha\) (Incorrect).
- **Option 2**: If \(|\vec{A}| > |\vec{B}|\), then \(\alpha < \beta\) (Correct).
- **Option 3**: If \(|\vec{A}| = |\vec{B}|\), then \(\alpha = \beta\) (Incorrect).

### Conclusion
From our analysis, we can conclude that the correct relationship is:
- If \(|\vec{A}| > |\vec{B}|\), then \(\alpha < \beta\).